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Parameter scaling in the decoherent quantum-classical

transition for chaotic systems

Arjendu K. Pattanayak(a) and Bala Sundaram(b)

(a) Department of Physics, Carleton College, Northfield, Minnesota 55057

(b) Graduate Faculty in Physics & Department of Mathematics,

CSI-CUNY, Staten Island, New York 10314

(Dated: February 1, 2008)

Abstract

The quantum to classical transition has been shown to depend on a number of parameters. Key

among these are a scale length for the action, h, a measure of the coupling between a system and

its environment, D, and, for chaotic systems, the classical Lyapunov exponent, λ. We propose

computing a measure, reflecting the proximity of quantum and classical evolutions, as a multivari-

ate function of (h, λ,D) and searching for transformations that collapse this hyper-surface into a

function of a composite parameter ζ = hαλβDγ . We report results for the quantum Cat Map,

showing extremely accurate scaling behavior over a wide range of parameters and suggest that, in

general, the technique may be effective in constructing universality classes in this transition.

PACS numbers: PACS numbers: 05.45.Mt,03.65.Sq,03.65.Bz,65.50.+m

1

The classical description of a system approximates the inherently quantum world and

has significantly different predictions. The question of when quantum mechanics reduces

to classical behavior is both fundamentally interesting as well as relevant to applications

such as quantum computing which seek to exploit this difference. The quantum to classical

transition (QCT) is now understood to be affected not only by the relative size of h (Planck’s

constant) for a given system, but also by D, a measure of the coupling of the environment to

the quantum system of interest, an effect termed decoherence. Further, in systems where the

classical evolution is chaotic, the transition is also affected by the chaos in the system, and

thus by λ, the Lyapunov exponent of the classical trajectory dynamics [1, 2]. As such, the

QCT for chaotic Hamiltonians is, in general, a complicated function of multiple parameters,

and is far from being fully understood.

However, the parametric dependence is as daunting as it first appears, particularly near

the transition regime. Several studies point to composite parameters, indicating that the

transition is not independently affected by each of the three parameters. For example,

considerations [1, 2, 3, 4, 5, 6] of stochastic quantum evolution or a master equation show

that the parameter range for classical behavior is not simply h ≪ 1 but depends also on D.

These and similar studies also indicate scaling relationships involving h, D, λ. Other work

has addressed correspondence at level of trajectories which requires a continuous extraction

of information from the environment [7], as opposed to tracing over these variables. However,

here again, the condition for correspondence may be viewed as a composite variable where

D is appropriately replaced by the strength of the measurement. More recently, it has been

argued that Hamiltonian systems fall into a range of universality classes with distinctly

different QCTs [8], behavior manifested in the density matrix far from the transition regime.

With these as motivation, we propose that significant progress can be made by (a) com-

puting measures which directly reflect the ‘distance’ between quantum and classical evolu-

tions as a function of h, λ, and D and then (b) searching for transformations that collapse the

resulting hyper-surface onto a function of a composite parameter of the form ζ = hαλβDγ.

The aims are (i) to search for this scaling, especially the coefficients α, β, γ [9]; (ii) to in-

vestigate the range of parameters and initial conditions over which the scaling holds and

(iii) to study the dependence of the distance measure on ζ . We can anticipate the possible

outcomes: First, that α, β, γ are independent of the Hamiltonian. If this extremely unlikely

scenario holds, we have a modified Planck’s constant governing all quantum chaotic systems,

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and universality classes are differentiated by differing dependences of the distance measure

on ζ . Second, a range of behavior for α, β, γ is seen, including a dependence on initial con-

ditions, providing a classification scheme possibly correlated with the previously proposed

classes. Finally, any scaling may be associated with the nature (single-scale, multi-scale) of

the quantum coherence affected by the environment. This suggests a third alternative where

scaling behavior exists only for limited classes of systems or limited parameter ranges, in

which case the existence or range of scaling defines universality classes.

Below, we present broad arguments for the existence of such scaling. We then consider

two alternate measures of the quantum-classical distance including a generalized Kullback

distance [10]. We numerically test our ideas with these measures on a specific system, the

noisy quantum Cat Map. For the Cat Map, the Lyapunov exponent is a constant, such that

the QCT is at most a two-parameter transition. We show that this two-parameter transition,

in fact, reduces to an effective single-parameter transition. This scaling is remarkably sharp

and extends over a large range of parameters. In the case of the Cat Map, the quantum

nature of the system is a well-defined function of ζ ≡ h2λD−1, consistent with previous

analysis [5]. We discuss the nature of the transition in some detail, and conclude with

expectations for the decoherent QCT in other, more general, chaotic systems.

We begin from the equation describing the evolution of a quantum Wigner quasi-

probability ρW under Hamiltonian flow with potential V (q) while coupled to an external

environment [1]:

∂ρW

∂t= {H, ρW} +

∑

n≥1

h2n(−1)n

22n(2n + 1)!

∂2n+1V (q)

∂q2n+1

∂2n+1ρW

∂p2n+1

+ D∇2ρW . (1)

The first term on the right is the Poisson bracket, generating the classical evolution for ρW .

The terms in h add the quantal evolution while the effects of the environmental coupling

are reflected in the diffusive term. For simplicity, we couple to all phase-space variables,

although the results generalize. Consider for the moment only the classical evolution in the

presence of the environmental perturbation. As a result of chaos, the density ρ develops fine-

scale structure exponentially rapidly, with a rate given by a generalized Lyapunov exponent.

When the structure gets to sufficiently fine scales, the noise becomes important. The basic

role of noise is to wipe out, or coarse-grain, small-scale structure. The competition between

chaos and noise leads to a metastable balance for the fine-scale structure [11]. This is clearly

3

visible in the measure χ2 ≡ Tr[ρW∇2ρW ]Tr[(ρW )2]

= −Tr[|∇ρW |2]Tr[(ρW )2]

where the second equality results from

an integration by parts. This quantity χ2 is approximately the mean-square radius of the

Fourier expansion of ρ and, for our purposes, measures the structure in the distribution [12].

For a classically chaotic system under the influence of noise, χ2 settles after a transient to

the metastable value χ2∗ =∑

i Λ+2,i/2D ≡ Λ/2D where the Λ+

2,i are ρ dependent versions of

the usual generalized positive Lyapunov exponents of second order [10, 11].

Now let us add quantal corrections to the mix. As seen from Eq. (1), the terms are

of the form h2n ∂2n+1V (q)∂q2n+1

∂2n+1ρW

∂p2n+1 which scale as h2nχ2n+1V (2n+1)(x), where V (r) denotes

the rth derivative of V . Since χ2 settles to the fixed value Λ/2D, this contribution

to the difference between the quantum and classical evolution may be estimated to be

ζ ≡ h2nΛn+1/2D−(n+1/2)V (2n+1)(x) where x ≈ χ−1 =√

D/Λ. Therefore, quantum-classical

distances should scale, in complete generality, with the single parameter ζ for small ζ . The

particular form of ζ is decided by the details of the Hamiltonian and, in general, the scaling

relationship is deduced from a direct examination of the deviation of the quantal propagator

from the classical version.

As a measure of the distance between two distributions P and Q with support on the

same space, we introduce the quantity

Kǫ(P, Q) =1

ǫ[ln(Tr[PQǫ]) − ln(Tr[P 1+ǫ])

+ ln(Tr[P ǫQ]) − ln(Tr[Q1+ǫ])] (2)

where Tr denotes the trace over all variables. Kǫ is a generalized Kullback-Liebler (K-L)

distance, reducing to a symmetrized form of the usual K-L distance [10] in the limit ǫ → 0.

To see this, use that P ǫ = exp(ǫ ln P ) ≈ 1 + ǫ ln(P ) + O(ǫ2) for ǫ → 0. Then, the first

term in Eq. (2) becomes 1ǫln(Tr[PQǫ]) ≈ 1

ǫln(1 + ǫTr[P ln Q]). Now using the expansion for

ln(1 + x) for small x for this and the other terms, this yields

limǫ→0

Kǫ(P, Q) = Tr[P ln(Q

P)] + Tr[Q ln(

P

Q)] (3)

which is indeed a symmetrized version of the usual K-L distance. Kǫ has similar properties,

and is a general measure of the distance between the two probability distributions. When

P and Q are identical, this measure is zero. A convenient form of Kǫ is for ǫ = 1 when it

reduces to

K1(P, Q) = ln[

(Tr[PQ])2

Tr[P 2]Tr[Q2]

]

. (4)

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We begin from an initial phase-space distribution ρ0, which is propagated in time using

separately (i) the quantum dynamics to yield ρW (t) and (ii) the classical dynamics for

ρc(t). During the propagation, the distance K1(ρW , ρc) is monitored. The initial distance

K1(t = 0) = 0, and due to diffusive noise all initial distributions relax to the constant

distribution, such that K1(t → ∞) = 0 and hence K1 is bounded as a function of time. For

a given set of parameters h, D and for some reasonably long time tm(≫ 1/Λ), the maximal

value of Km1 (ρW , ρc) is our measure of the quantum-classical distance.

0.5

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1.5

2

2.5

3

3.5

4

4.5

5

−3.9−3.4

−2.9−2.4

−1.9

−5.5−5

−4.5−4

−3.5−3−2

0

2

4

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PSfrag repla ementslog �hlogD

Km 1

FIG. 1: Top: Maximal Kullback-Liebler distance Km1 as a function of h and D, for the Quantum

Cat Map. Note that small values reflect strong similarity between classical and quantum evolutions.

Bottom: Same data plotted in terms of a composite parameter reflecting scaling behavior.

We illustrate the technique by considering a simple but extensively studied system, the

noisy quantum Cat Map [2, 5, 13]. The classical limit displays extreme (uniformly hyper-

bolic) chaos, and as such the system should be a member of a distinct universality class. The

uniform hyperbolicity also precludes any dependence on initial conditions. The dynamics

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derive from the kicked oscillator Hamiltonian [14]

H = p2/2µ + ǫq2/2∞∑

s=−∞

δ(s − t/T ). (5)

restricted to the torus 0 ≤ q < a, 0 ≤ p < b, with the parameter constraints Tb/µa = 1

and −ǫTa/b = 1. The chaos here results not from the non-linearity of the Hamiltonian but

from the choice of (re-injected) boundary conditions. As such, the general equation Eq. (1)

does not apply. However, the first quantum correction to the classical propagator for this

system (for the the Fourier-transformed distribution) is of order hk for the Fourier mode

k [13]. The quantum-classical distance for this system then behaves as hχ, implying that [15]

ζ = h2χ2 = h2ΛD−1. The top panel of Fig. 1 shows Km1 as a function of h, D. It is clear the

distance behaves as expected. For example, as h is increased, larger D values are needed

for the quantum and classical distributions to coincide. The lower panel shows the same

data, plotted as a function of the single composite variable ζ = h2/D. The reduction of the

surface in the upper panel to a single function of ζ demonstrates the scaling relationship

between h, D. The accuracy of this scaling is reflected in the lack of any discernible spread

around the curve. Remarkably, the scaling extends over many orders of magnitude in both

parameters h, D and a considerable range in Km1 .

The functional dependence of Km1 on ζ shows a number of distinctive features. (i) Km

1

is monotonic in ζ , although as we argue below, there is no general reason to expect this.

(ii) The quantum-classical distance is nonlinear in ζ , with Km1 (ζ) initially growing slowly

as a function of ζ , followed by a rapid transition at ln(ζ) ≈ 0 or ζ ≈ 1. This boundary

is consistent with previous results [2, 5, 6]. (iii) The distance K1 is bounded due to the

noise, and we see the expected saturation for higher values of ζ . (iv) There appear to

be distinct regimes corresponding to small (for ζ < 1) and large (for ζ > 1) quantum-

classical distance. This last behavior is arguably generic as, in chaotic systems, a classical

distribution develops fine-scaled structure very quickly (χ2 grows rapidly), increasing its

entropy production rate as well as its sensitivity to external noise. For this class of systems,

in the first regime (ζ < 1), a quantum distribution initially remains close to the classical

and will also increase its entropy production rate, and consequently the rate at which it

becomes a mixed state. Hence any quantum effects that develop will be suppressed by the

noise and the quantum-classical distance will remain small for all times. In this regime, the

environment minimizes the quantum-classical difference. In the second regime (for ζ > 1),

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0.1

0.2

0.3

0.4

0.5

0.6

−3.9−3.4

−2.9−2.4

−1.9

−5.5−5

−4.5−4

−3.5−3

−0.5

0

0.5

1

PSfrag repla ementslog �hlogD

D�2

FIG. 2: Top: This measure reflects the generation of fine-scale structure in the dynamics with larger

values corresponding to classical dynamics. Bottom: Same data plotted in terms of a composite

parameter. Note the same scaling as in Fig 1 and the coincidence of the transition region.

the quantum distribution does not initially follow the classical distribution to finer scales, and

does not become sensitive to noise. It thus remains far from classical even as the noise alters

the classical system. Here, the environment exaggerates the differences between quantum

and classical probability dynamics. As such, ζ ≈ 1 may be viewed as a ‘quantum-classical

boundary’, with qualitatively different behavior on either side of it.

The general arguments above imply that similar scaling should be visible in all appro-

priately constructed measures of the quantum-classical distance. In Fig. 2 we show results

for an alternate measure Dχ2, which is related to the spreading of structure to finer scales.

Unlike Km1 which compares classical and quantum evolution, this second measure is strictly

quantum mechanical. The supremum value in time of Dχ2 (≡ Dχ2m) is considered with vary-

ing h, D and for the same time-scales as before (classically, we would get a constant [11]).

Again, the precision and range of the scaling is remarkable. The qualitative conclusions

are exactly the same as for K1m(ζ), with a similar rapid transition between large and small

7

values of Dχ2m, happening again at ζ ≈ 1. That is, for small ζ , the distribution is very

sensitive to noise, changing rapidly as a function of ζ to low sensitivity. However, this curve

has a distinctive dip near ζ ≈ 1, such that the peak is at finite ζ . This has been seen pre-

viously [12], and can be understood by the fact that for near-classical quantum dynamics,

the quantum follows the classical distribution but carries interference fringes on top of the

classical structure. As such, the quantum distribution can be more sensitive to noise than

the classical counterpart. In particular, as above, ρW ≈ ρc +ahχρ where a is some constant.

Similarly, the quantum and classical χ2 are related as χ2q ≈ χ2

c +ahχ3 so that to zeroth order

χ2q0 = χ2

c , where the subscript on χq indicates the order. To first order, we substitute the

zeroth order expression for χ3 to get χ2q1 ≈ χ2

c + ahχ3c . Iterating this procedure, to second

order we will get terms like [16] χ2q2 ≈ χ2

c + ahχ3c(1 + ahχc)

3

2 . For small ah this becomes

χ2q ≈ χ2

c(1 + a hχc +3

2a2 h2χ2

c +3

8a3 h3χ3

c + . . .)

≈ χ2c(1 + a′ ζ

1

2 + b ζ + c ζ3

2 + . . .) (6)

where the constants a′, b, c absorb all other constants and we have substituted h2χ2 = ζ .

The initial effect of quantum dynamics is to reduce the value of χ2 and hence a′ (and

consequently c) must be negative valued constants, while b is positive. For appropriate

values of a′, b, c, Eq. (6) can indeed account for the shape of the curve seen in Fig. (2).

Therefore, all measures of quantum-classical distance need not depend monotonically on

the system parameters. However, the particular dependence shown is almost definitely not

generic since it depends on the relevant constants being of the appropriate ratios.

These results provide definitive evidence of parameter scaling in QCT for chaotic systems,

which may be used to clearly identify different regimes of quantum-classical correspondence.

As such, these are the first steps towards identifying and using composite parameters in

studying universal behavior in the quantum-classical transition for small ζ (the near-classical

regime). The smoothness and breadth of the scaling results shown are likely to be a feature

of the uniform hyperbolicity of systems like the Cat Map. Understanding how this is altered

by less extreme dynamics is clearly the next step, both in terms of constructing ζ and

well as exploring the dependences of computed measures on ζ . In particular, a preliminary

assessment of entirely different measures applied to the quantum Duffing problem indicates

that similar scaling may exist there as well.

Acknowledgement - A.K.P. acknowledges with pleasure useful comments from Doron Co-

8

hen and Ivan Deutsch. The work of B.S. was supported by the National Science Foundation

grant #0099431 and a grant from the City University of New York PSC-CUNY Research

Award Program.

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[8] S. Habib, K. Jacobs, H. Mabuchi, R. Ryne, K. Shizume, and B. Sundaram, Phys. Rev. Lett.

88, 040402 (2002).

[9] A dependence of the QCT on ζ implies such a scaling dependence exists in the parameter ζr

as well, for arbitrary r. We may choose one coefficient (e.g., setting α = 2) to determine the

other two coefficients.

[10] C. Beck and F. Schlogl, Thermodynamics of chaotic systems, (Cambridge University Press,

N.Y., 1993).

[11] A.K. Pattanayak, Physica D 148, 1 (2001).

[12] Yuan Gu, Phys.Lett.A 149, 95 (1990); A. K. Pattanayak and P. Brumer, Phys. Rev. E 56,

5174 (1997).

[13] J. Wilkie, Ph.D. Dissertation (unpublished), University of Toronto, 1994.

[14] J.Ford, G.Mantica and G.H.Ristow, Physica D 50, 493 (1991).

[15] We set ζ = hαΛβDγ , and choose α = 2 as discussed in [9] above. Our figure has a logarithmic

scale for ζ, whence this choice affects only the aspect ratio of the figure.

[16] In general, the expansion for χq around χc in terms of ζ includes terms in ζ arising from

the higher-order terms in the relationship between ρW and ρc. For the argument that the

relationship between χq and χc is nonlinear in ζ, these may be neglected since they only add

to the nonlinearity.

9